Equilibrium fluctuations in relativistic thermoelectric fluids

A NNALES DE L ’I. H. P., SECTION A

D. P ^AVÓN D. J ^OU

J. C ^ASAS -V ^ÁZQUEZ

Equilibrium fluctuations in relativistic thermoelectric fluids

Annales de l’I. H. P., section A, tome 42, n

^o

1 (1985), p. 31-38

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31

Equilibrium fluctuations in relativistic thermoelectric ^fluids

D.

PAVÓN,

^D. JOU and J.

CASAS-VÁZQUEZ

Departamento ^de Termologia. Facultad de Ciencias.

Universidad Aut6noma de Barcelona. Bellaterra. Barcelona (Spain)

Vol. 42, ^n° 1, 1985, Physique theorique

ABSTRACT. 2014 The formalism of the extended

thermodynamics

^is

applied

to a relativistic thermoelectric fluid. Two new terms appear in the genera- lised Gibbs

equation.

^Their

physical meaning

^{is studied}

by

^{means of the}

analysis

^{of the} fluctuations of the

dissipative

fluxes. The constitutive _equa- tions as well as the

fluctuation-dissipation

^{theorem are obtained.}

RESUME. ^- On

applique

Ie formalisme d’une

thermodynamique gene-

ralisee

(extended thermodynamics)

^{a un fluide}

thermoelectrique

relativiste.

Nous

soulignons

^la

presence

^{de deux nouveaux termes dans}

1’equation

de Gibbs

generalisee,

^{dont la}

signification physique

^{est mise en evidence}

a 1’aide de la theorie des fluctuations des fluides

dissipatifs.

^On deduit aussi Ie theoreme de

fluctuation-dissipation

^{et les}

equations

constitutives du fluide.

1 INTRODUCTION

For many years the role

played by

the relaxation _proper times in the

description

^of

dissipative phenomena

^{has not been}

properly

^{taken into}

account.

Recently

^however

causality requirements

have thrown them into an

outstanding

^{level in classical}

[7] as

^{well as} in relativistic

[2] ] [3]

formulations of these

phenomena. Usually

the mentioned _proper times

are introduced into the

theory through

^a

non-equilibrium

entropy

function,

Annales de l’Institut Henri Poincaré -

Physique

theorique - ^{Vol. 42, 0246-0211} 85/01/31/ 8 /$ 2,80/(0 Gauthier-Villars 2

32 D. PAVON, D. JOU AND J. CASAS-VAZQUEZ

whose

physical meaning

^{in some}

specific

situations has been

explored by

^{two of us}

[4] ] [5 ].

In this note we

analyse

^the simultaneous _process of heat and electric conduction in a relativistic inviscid fluid as well as the fluctuations of both fluxes around

equilibrium.

^{In our}

study

the relaxation _proper times of the process under consideration

play a

^{main role. Let us}

imagine

for instance

a thermoelectric fluid submitted to a

temperature gradient ; consequently

a heat and an electric flux will _appear. If the temperature difference res-

ponsible

^{for the}

temperature gradient

^is

suddenly removed,

both fiuxes will not vanish

immediately

^{but after a} finite time. This reflection suggests that the relaxation _proper times must enter on their own

right

^{in the ther-}

modynamical description

of the involved _processes.

We start from the relativistic version of extended irreversible thermo-

dynamics [3 ].

^The

ensuing

constitutive

equations

^{reduce to the earlier}

obtained

by

^{us in the} limits of non heat

[6] and

^{non electric}

[7] conducting

fluids

respectively.

Likewise in both limits the second moments for the fluctuations of the

dissipative

^fluxes go into the

expressions

^{derived in a}

previous

paper

[8 ].

The outline of this note is as it follows. In Section 2 we present ^the

phe- nomenological description

^{of the} relativistic thermoelectric fluid

including

besides a relation

holding

between the different relaxation _proper times.

In Section 3 we

analyse

^the

equilibrium

fluctuations of

dissipative

^fluxes.

Lastly,

^{Section 4} is devoted to some final remarks.

As it is customary 0"w denotes the

spatial projector

⁺

MV,

^with

~

the metric tensor of

signature (2014,+,+,+)

^and u"~ the world

velocity

normalized

according

^{to 1.} Derivation

along

^u"~ will be indicated

by

^{means of en} upper dot.

2. RELATIVISTIC DESCRIPTION OF THERMOELECTRIC FLUIDS

Let us assume a relativistic inviscid fluid

capable

of heat and electric

conduction,

^{submitted to an external}

electromagnetic

^{field and with}

a momentum-energy ^tensor

given by

where

q"~

stands for the heat flux and 8 for the internal

specific

^energy.

The evolution of this latter

quantity along

the world line is afforded

by

the balance

equation

v

being

^{£ the}

specific

^{volume ’}

(v

⁼

1/p),

^{and ’ 1~} the electric conduction current.

Annales de l’Institut Henri Poincaré - Physique ^’ theorique ^’

By

^{virtue of the} conservation of the electric

charge,

^{this vector}

obeys

^to

the relation

z

being

^the

specific

^electric

charge,

^which

yields

^the evolution of this _quan-

tity along

Moreover Iu shares

with q

^the well-known

geometric

pro- perty ^{= =}

0 ;

^{i. e. both vectors are of}

spatial

type. ^{As we will}

see, this fact makes _easy the mathematical treatment of our

problem.

Following

the lines of the extended relativistic

thermodynamics

^we

assume that the

dissipative quantities

^enter besides the

equilibrium

^variables

in the

non-equilibrium specific

entropy

function ~

^{as well as in the entropy}

flux

1~). ^Thus,

^we

^postulate

^{on the one hand the}

generalized

^entropy

r~ ⁼

r~(E, v,

^z,

P)

whose Gibbs

equation

^can be written as

and on the other

hand,

^the

following expression

^{for the} entropy ^{flux vector}

These two

expressions

^{are the most}

general

^{that can} be constructed _up to second order in the

dissipative

fluxes for an

isotropic

^{fluid. The chemical-}

like

potential

,ue is defined

by

^{means of the}

equation

^{of state}

(~h/~z)’ = - e/T,

where an upper

prime

^{denotes that all}

quantities

^{but z are to be}

kept

^constant

during

^the derivation. The coefficients aij ^as well as

the 03B2i (i, j = 1, 2)

are functions of _{E, v} and _{z ;} the former

being

relaxation

parameters

^which will be identified later.

Comparison

^of

(5)

with the classical

expressions

enables us to

identify /31

^and

/32

^as

1/cT

^{and -}

,ue/T respectively. Since ~

must be a

perfect differential,

^{the Maxwell relation} a12 ⁼ a21 is satisfied.

Also,

^the

following

^set of restrictions exists on the ai~

which arise from the

requirement of ~ being

^{maximum at}

equilibrium

^state.

By combining

^the entropy ^balance

equation p~

⁺

1~

^{= (7 with}

(2)-(5)

we set for the entronv nroduction

with ⁼

A~X~,

⁼ and where the

generalized

^thermo-

dynamical

^{forces are}

given according

^to

respectively.

Vol. 42, ^{n° 1-1985.}

34 D. PAVON, ^{D. JOU AND J.} CASAS-VAZQUEZ

In order to get ^the

transport equations

^we

expand

^both

conjugate

^forces

in function of the

dissipative

^{’ fluxes} up ^to second ^, order in these ^"

variables;

thus one ^" has

where the coefficients

a~~ (i, j

⁼

1, 2) depend

^{on the}

equilibrium

^variables

only,

^and

they

^are also submitted to the restrictions

which

proceed

^{from the}

semipositive

definite character of cr, ^as it _may

readily

^checked

by inserting ( 10)

^and

( 11 )

^into

(7).

^From

equations (8)-( 11 )

the

henomenoloical

^relations

arise.

They

^{form a set of}

equations giving

^the evolution of the

dissipative

fluxes

along

the world line of each

particle

^{mass element. In this} way

(13)

and

( 14) together (2)

^and

(3)

^{and the}

continuity equation

^{= 0}

describe the behaviour of the fluid as it evolves

along Obviously (13)

and

(14)

^recover

respectively

^the transport

equations already

^{deduced in}

two

previous

papers

[7] ] [6 ].

It remains to determine the parameters _ai~ ^{as well as the This can} be done

by especializing ( 13)

^and

( 14)

^{at the}

comoving

frame 2014 in it one

has

A~

⁼

diag (0, 1, 1, 1 )

^{- then}

by comparing them,

^{in the}

stationary

case, with the well-known

equations

^{of the} classical literature

[9 ].

^Thus

we get

were and _X stand for the heat and the electric

conductivity

of the fluid

respectively,

^{whereas ~f and () are}

respectively

the differential thermo- electric _power and the heat at uniform

temperature

per unit electric current.

Once related the a~~ ^to measurable

quantities

^the ai~ ^can be obtained if

(13)

and

( 14)

^are

compared, again

^{in the}

comoving frame,

and for the non-

stationary

^case, with the relaxed classical

equations.

^This

gives

^{rise to}

Here

0)

^are the relaxation _proper times of the involved _processes.

The afore-mentioned Maxwell

relation,

a 12 - a21,

together

^{with the}

Annales de l’Institut Henri Poincare - Physique theorique

second and third

equations

^of

(16), implies

⁼

i21(0

⁺

Moreover,

^{if the}

Onsager

^relation a12 ⁼ a21 is satisfied it follows the

more

stringent

^condition ~2=~21.

Actually,

^its

validity

^{can be assured}

in the

equilibrium only

^{since the}

Onsager

^relations may ^not hold outside

equilibrium [10 ].

At any rate, ^there exists another restriction on the ii~

which

proceeds

^{from the third}

equation

^of

(6)

and it reads

In some circumstances _03C411 and _{L 22 may} be

measured,

^{and then the last}

inequality

^{would be}

employed

^{to set upper limits to both 03C412 and i21.}

- 3. FLUCTUATIONS

OF THE FLUXES AROUND

EQUILIBRIUM

As it is

well-known,

^{in a} thermoelectric fluid heat fluctuations

originate

electric current fluctuations and vice versa.

Therefore,

ⁱⁿ

dealing

^{with such}

a medium both kind of fluctuations must be considered

together.

^{On the}

other

hand,

^{since both of them are}

mutually implied,

it is reasonable to

expect

^a

non-vanishing

correlation amongst ^them.

Let us assume a thermoelectric fluid at

equilibrium.

^{In such a case we}

have ⁼

0,

^{and the}

average

^values

of qu

^{and 1~}

^vanish,

but however both vectors can fluctuate around that value. In addition the acceleration must vanish also since

otherwise,

^{as a} consequence of the inertia of

heat,

a heat flux would arise. In order to obtain the

probability

^{of a}

given

spon- taneous fluctuation or we resort to the Einstein-Boltzmann _expres- sion

largely

^{used in the literature}

where M

and kB

^{stand for the mass of the} system and the Boltzmann constant

respectively. By combining

^the

generalized

^Gibbs

equation (4)

^with

( 18)

it follows

and from this last

expression

^{the second moments} calculated in a

given

event

point,

^say

x~,

^read

Vol. 42, ^n° 1-1985.

36 D. PAVON, ^{D. JOU AND J.} CASAS-VAZQUEZ

oc*

being (03B11103B122

^-

a 12 a21 ) 1

^{and V the} volume of the system. ^These three

equations

^are relativistic versions of the

fluctuation-dissipation

theorem

relating

the second moment of fluctuations ^to the

phenomeno- logical

coefficients. Note that in the limit when _{L12 ~} 0

equations (20)

and

(22)

^{reduce to their} counterparts deduced in

[8] ] and,

^{on the other}

hand, All v [~ 51 ] -~

^0.

Since,

^on

physical grounds

this latter

quantity

must be different from _zero, relation

(21 )

may be

regarded

^{as an indirect}

corroboration of the _presence of the crossed terms

v03B112I 03B1q

^and

v03B121q I

in the

generalized

^Gibbs

equation. Notwithstanding,

it is convenient to remind that

(4)

^is

only

^an

approximation

up ^to second order in the dissi-

pative

^fluxes.

Because of the

spatial

character of and the restrictions

must be satisfied

by (19)

^{for a}

comoving

^{observer. Next we show that}

(19)

bears such a property.

Effectively,

^{for such an observer we have from}

(20)-(22)

the relations

A%[~]=A%[~,~I]=A%[(5I]=0,

^and as a conse-

quence

( 19)

^fulfills the above restrictions.

The correlation

functions, A~[~....],

^{for the} fluctuations between two very close event

points,

say x03C3 and x03C3 + which lie on the same world line can be obtained from

(20)-(22)

and the evolution

equation

^{for the}

fluctuations. These later follow from

(13)

^and

(14) respectively,

^and

they

^read

In

deducing

^{them we have}

neglected

^the fluctuations of the

gradients T,v

and since

they

fluctuate much slower than the heat flux and the electric current

density.

^{After some}

manipulations

^{the set of}

equations

becomes ^’

with the matrix M of the coefficients

Mi~

^defined

through

^{M =} 0152 - 1 . a.

Since the matrix IX - 1 is

negative

semidefinite and a is

positive semidefinite,

M is

negative

semidefinite.

Integration

of the above set of

equations

between x03C3 and x03C3 +

yields

where the

Ai~

^{terms are}

given by Ai~

⁼ exp ^s

being

^{the arc}

lenght

measured

along

the world line. The

negative

semidefinite character of M

Annales de l’Institut Henri Poincare - Physique theorique

guarantees ^the

decay

of both fluctuations and

Finally

^from

(20)-(22)

^and

(27) (28)

^we get

As it can be _seen, the second ^moments in two

separated

^event

points

^can

be

expressed,

^{in this}

simple

way, ^{as a} linear combination of the correla- tions

A~[5...].

4. CONCLUDING REMARKS The form of the

non-equilibrium

part ^{of the} entropy,

+ + +

of the

generalized

^Gibbs

equation (4)

^was

proposed,

ⁱⁿ

principle,

^{in basis}

to mathematical

requirements only. Later,

^{we found on the one hand the}

meaning

^{of the} 03B1ij and their

implication causality

^{2014 on the}

transport equations and,

^{on the other}

hand,

^we had observed that the cannot vanish since in that case the correlations

A~[5...] ]

would vanish

also,

which is

against physical

^sense.

Moreover,

^on

physical grounds,

^{we know}

that the relaxation _proper

times, although generally small,

^{are finite and}

hence the 03B1ij ^cannot be set to zero.

To

summarize,

^{we se that the enter not}

only

^{in the}

macroscopic

^des-

cription

of the irreversible _processes

(transport equations),

^{but also in}

the

mesoscopic

^one

(fluctuations).

^{Between both of them it exists a}

bridge, namely

^the

fluctuation-dissipation theorem,

^whose relativistic version for thermoelectric fluids has been

analysed

^{here in a} relaxation time

approxi-

mation.

ACKNOWLEDGMENTS

This work has been

partially supported by

^the Comision Ascsora de

Investigacion

Cientifica _y Tecnica of the

Spanish

Government and

by

NATO Research Grant n°

0355/83.

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[2] ^{W. ISRAEL and J. M.} STEWART, Transient relativistic thermodynamics and kinetic theory, ^Ann. Phys., ^t. 118, 1979, p. 341-372.

[3] ^D. PAVÓN, D. Jou and J. CASAS-VÁZQUEZ, ^{On a} covariant formulation of dissipa-

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Vol. 42, ^{n° 1-1985.}

38 D. PAVON, ^D. JOU AND J. CASAS-VAZQUEZ

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[8] ^D. PAVÓN, ^{D. Jou and} J. CASAS-VÁZQUEZ, ^Heat and electric fluctuations. A relativistic approach, ^J. Non-Equilib. Thrmodyn., ^t. 6, 1981, p. 173-180.

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(Manuscrit reçu Ie 25 janvier 1984)

de Poincaré - Physique theorique